Problem 41 - College Algebra

Question

Evaluate each expression without using a calculator. $$ 8^{\log _{8} 19} $$

Step-by-Step Solution

Answer

19

Step 1: Identify the Expression

We need to evaluate or simplify: \(8^{\log _{8} 19}\).

Step 2: Apply Order of Operations

Following the order of operations (PEMDAS/BODMAS), we simplify the expression step by step, handling parentheses, exponents, multiplication/division, then addition/subtraction.

Step 3: Compute the Result

Carrying out the operations systematically, we arrive at the result.

Step 4: State the Result

\(19\)

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions of the form \( f(x) = b^x \), where \( b \) is a positive real number, and \( x \) is any real number. This function grows rapidly: the higher the value of \( x \), the greater the value of \( f(x) \).

Imagine a scenario where you're multiplying a number repeatedly by itself; that's exactly what an exponential function represents. A classic example is compound interest in finance, where the amount of money grows exponentially over time depending on the rate of interest and the number of times it is compounded. In our exercise problem, \( 8^{\log_8 19} \) is an exponential expression where 8 is the base raised to the power of the logarithmic function \(\log_8 19\).

To understand such functions deeply, it's crucial to explore the base, which dictates how quickly the function will grow. If this base is greater than 1, the function grows as x increases. When the base is between 0 and 1, the function will decrease as x increases, known as exponential decay. Exponential growth and decay can be seen in various real-life applications including population growth, radioactive decay, and even certain algorithms in computer science.

Logarithmic Functions

Logarithmic functions are essentially the inverse of exponential functions and are written as \( g(x) = \log_b x \), where \( b \) is the base and \( x \) is the value we're taking the logarithm of. The base \( b \) is also a positive real number, and \( x \) must be positive as well. The purpose of the logarithm is to find the power to which the base must be raised to obtain \( x \).

Understanding Logarithms

In simpler terms, if we have \( b^y = x \), then \( y = \log_b x \). So, for the function \( 8^{\log_8 19} \), what we're really asking is to what power we must raise 8 to get 19. The answer to this is the value of \( \log_8 19 \), which is then used as the exponent for base 8. This relationship is a core property of logarithms, and it's extremely helpful in solving various algebraic and real-world problems. Logarithms are commonly used in science to describe phenomena observed on a huge scale of values, such as in chemistry with the pH scale or in astronomy when measuring the brightness of stars.

Evaluating Expressions Without a Calculator

Evaluating expressions like \( 8^{\log_8 19} \) without a calculator involves using fundamental properties of logarithms and exponents to simplify the expression. Ideally, to solve such logarithmic and exponential functions without a calculator, one needs a strong understanding of these properties.

Property Understanding:

• Laws of Exponents: These include the product rule, power rule, and quotient rule, which are crucial in manipulating expressions involving exponents.
• Logarithm Properties: Key properties such as \( b^{\log_b a} = a \), \( \log_b (b^x) = x \), and change-of-base formulas allow for simplification and conversion between forms.

We leveraged a very useful property in our exercise solution that when the base of the exponential and the base of the logarithm match, the two functions essentially 'cancel' each other out, leaving us with the argument of the logarithm. This property is what made it possible to evaluate \( 8^{\log_8 19} \) as simply 19. In real-world scenarios, being able to do these calculations by hand is not only practicable for understanding and troubleshooting complex problems but also vital in cases where technology might not be accessible.